Question

determine which of the lines, if any, are parallel or perpendicular. explain.
lines a and b are parallel. the slopes are the same .
lines c is perpendicular to lines a and b . the slopes are not reciprocals .

Explanation:

Step1: Calculate slope of line a

Points for line a: \((-1, 1)\) and \((2, 0)\). Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). So \(m_a=\frac{0 - 1}{2 - (-1)}=\frac{-1}{3}=-\frac{1}{3}\).

Step2: Calculate slope of line b

Points for line b: \((0, 5)\) and \((3, 4)\). Using slope formula: \(m_b=\frac{4 - 5}{3 - 0}=\frac{-1}{3}=-\frac{1}{3}\).

Step3: Calculate slope of line c

Points for line c: \((0, 0)\) and \((2, 5)\). Slope: \(m_c=\frac{5 - 0}{2 - 0}=\frac{5}{2}\).

Step4: Analyze parallel/ perpendicular

• Parallel: Lines with equal slopes. \(m_a = m_b=-\frac{1}{3}\), so a and b are parallel.
• Perpendicular: Product of slopes is \(-1\). \(m_a\times m_c=-\frac{1}{3}\times\frac{5}{2}=-\frac{5}{6}

eq - 1\), \(m_b\times m_c=-\frac{1}{3}\times\frac{5}{2}=-\frac{5}{6}
eq - 1\). So c is not perpendicular to a or b.

Answer:

Lines a and b are parallel (slopes \(m_a = m_b=-\frac{1}{3}\)). Line c is not perpendicular to a or b (product of slopes with a/b is \(-\frac{5}{6}
eq - 1\)).